All of the 5th grade teachers and students from Santa Rita went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$2.00$ each for students, and the group paid $$41.50$ in total. The next month, the same group visited a science museum where the tickets cost $$34.00$ each for teachers and $$9.50$ each for students, and the group paid $$178.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2y = 41.5}$ ${34x+9.5y = 178}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-8y = -166}$ ${34x+9.5y = 178}$ Add the top and bottom equations together. $ 1.5y = 12 $ $ y = \dfrac{12}{1.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {8.5x+2y = 41.5}$ to find $x$ ${8.5x + 2}{(8)}{= 41.5}$ $8.5x+16 = 41.5$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {34x+9.5y = 178}$ and get the same answer for $x$ ${34x + 9.5}{(8)}{= 178}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.